Course Staff

Course Timetable

Course Learning Outcomes

In 2020, the topic of this course is Differential Geometry.

Syllabus

This course is concerned with the generalisation of multivariable calculus to settings more general than Euclidean spaces. It provides the foundation for advanced studies in analytical mathematics and physics, amongst other fields. The topics covered include a review of multivariable calculus in Euclidean spaces; manifolds; differential forms; the general form of Stokes' theorem, line bundles; connections, curvature and the chern classes of a line bundle; and de Rham cohomology of manifolds.

Assumed knowledge for the course is some form of multivariable calculus and a working knowledge of linear algebra.

Learning Outcomes

On successful completion of this course, students will be able to

1. define and recognise a differentiable manifold, and perform calculations on them;2. differentiate, integrate and pull back differential forms on manifolds;3. state and apply the general form of Stokes' theorem;4. recognise line bundles on manifolds and construct connections on these;5. calculate the curvature of a connection, and explain the relationship between curvature and the chern class of the line bundle;6. define and use de Rham cohomology groups of a manifold, and calculate these in simple cases.

University Graduate Attributes

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute

Course Learning Outcome(s)

Deep discipline knowledge

informed and infused by cutting edge research, scaffolded throughout their program of studies

acquired from personal interaction with research active educators, from year 1

accredited or validated against national or international standards (for relevant programs)

all

Critical thinking and problem solving

steeped in research methods and rigor

based on empirical evidence and the scientific approach to knowledge development

Required Resources

Lecture notes will be provided.

Recommended Resources

There are many excellent resources on differential geometry available including books which can be downloaded from the Barr Smith Library and other lecturers notes on the internet. The following is a short selection of some that are compatible with the objectives and the level of this course:

Online Learning

Learning & Teaching Modes

The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

Workload

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

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